Modern Engineering Thermodynamics

11.10 Compressibility Factor and Generalized Charts 385
dimensionless equation of state. He defined the
dimensionless variables reduced pressure p R , reduced
temperature T R ,andreduced specific volume v r as 4
p R = p/p c
(11.38a)
van der Waals
isotherms
T R = T/T c
v R = v/v c
(11.38b)
(11.38c)
where p and T are the actual pressure and temperature of
the gas, and p c and T c are the critical state pressure and
temperature of the gas (see Table C.12 in Thermodynamic
Tables to accompany Modern Engineering Thermodynamics).
Then, he hypothesized that p R = p R (v R , T R ) would define
a generalized dimensionless equation of state that
would be valid for all substances. Today, this hypothesis
is called van der Waals’s law of corresponding states, but
unfortunately, it has been found to be valid only for
materials with similar molecular structures.
In 1883, van der Waals introduced his now classical equation of state:
p = RT/ðv − bÞ − a/v 2 (3.44)
in which the constants a and b are corrections for intermolecular forces and molecular volume, respectively. This
is a cubic equation in v (see Example 11.11) and has isotherms shaped as shown in Figure 11.3. For given
p and T < T c values, there are three real roots of this equation. One root corresponds to v f , another corresponds
to v g , and the third is a meaningless root between v f and v g .
Van der Waals noted that the critical temperature isotherm seemed to have an inflection point at the critical
point. If this were generally true, then all equations of state would have to obey the mathematical constraints of
an inflection point, or
∂p
=
∂v
Tc,v ∂2 p
c ∂v
Tc,v 2 = 0
c
Applying these conditions to the van der Waals equation yields
and
∂p Tc
= RT c / ðv c − bÞ 2 + 2a/vc 3 ∂v
= 0
,v c
∂ 2
p Tc
∂v 2 = 2RT c / ðv c − bÞ 3 − 6a/vc 4 = 0
,v c
and solving these two equations simultaneously for a and b while using Eq. (3.44) evaluated at the critical point
gives
and
a = 9RT c v c /8 = 27
R 2 Tc
2
64 p c
b = v c /3 = RT c
8p c
Specific volume, v
Thus, if the van der Waals equation accurately represented universal material behavior, the constants a and b
could be determined from a single experimental measurement at the critical point.
Pressure, p
Vapor dome
Actual
isotherm
FIGURE 11.3
Schematic of van der Waals isotherms near the vapor dome.
4 Note that the reduced properties p R and v R are not the same as the relative properties p r and v r introduced in the previous section.